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We consider the component structure of the random digraph D(n,p) inside the critical window 
$p = n^{-1} + \lambda n^{-4/3}$
. We show that the largest component 
$\mathcal{C}_1$
has size of order 
$n^{1/3}$
in this range. In particular we give explicit bounds on the tail probabilities of 
$|\mathcal{C}_1|n^{-1/3}$
.
